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University of Toronto, November 16, 2004

This document in PDF: Exam.pdf

**Solve 5 of the following 6 problems. ** Each problem is
worth 20 points. If you solve more than 5 problems indicate very
clearly which ones you want graded; otherwise a random one will be left
out at grading and it may be your best one! You have an hour and 50
minutes. No outside material other than stationary is allowed.

**Problem 1. ** Let be an arbitrary
topological space. Show that the diagonal
,
taken with the topology induced from , is homeomorphic to
. (18 points for any correct solution. 20 points for a correct
solution that does not mention the words ``inverse image'', ``open
set'', ``closed set'' and/or ``neighborhood''.)

**Problem 2. ** Let be a connected metric
space and let and be two different points of .

- Prove that if then the sphere of radius around , , is non-empty.
- Prove that the cardinality of is at least as big as the continuum: .

**Problem 3. **

- Define `` is completely regular''.
- Prove that a topological space can be embedded in a cube (a space of the form , for some ) iff it is completely regular.

**Problem 4. **

- Define `` is (normal)''.
- For the purpose of this problem, we say that a topological space is if whenever and are disjoint closed subsets of , there exist open sets and in so that , and . Prove that if is then it is also .

**Problem 5. ** The ``diameter'' of a metric space
is defined to be
.

- Prove that a compact metric space has a finite diameter.
- Prove that if is a compact metric space, then there's a pair of points so that .

**Problem 6. ** If is a sequence of continuous
functions
such that
for each
,
show that is continuous at uncountably many points of
.

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Dror Bar-Natan 2004-11-16